Intro to triangle similarity

when we compare triangle ABC to triangle XYZ it's pretty clear that they aren't congruent that they have a very different lengths of their sides but there does seem to be something interesting about the relationship between these two triangles one all of their corresponding angles are the same so the angle right here angle BAC is congruent to angle yxz angle BCA is congruent to angle y ZX and angle ABC is congruent to angle XYZ so all of their angles they're the corresponding angles are the same and we also see we also see that the sides are just scaled up versions of either of each other so to go from the length of XZ to AC we can we can multiply by 3 we multiplied by 3 there to go from to go from XY the length of XY to the length of a B which is the corresponding side we are multiplying by 3 we have to multiply by 3 and then to go from the length of Y Z to the length of BC we also we also multiplied we also multiplied by 3 so essentially triangle ABC is just a scaled up version of triangle XYZ if they were the same scale they would be the exact same triangles but that one is just a bigger a blown-up version of the other one or this is a miniaturized version of that one over there if you just multiply all the sides by 3 you get to this triangle and so we can't call them congruent but there seem to be a bit of a special relationship so we call the special relationship similarity so we can write that triangle triangle a bc is similar similar to triangle and we want to make sure we get the corresponding sides right ABC is going to be similar to X Y Z to X Y Z and so based on what we just saw there's actually kind of three ideas here and they're all equivalent ways of thinking about similarity one way to think about it is that one is a scaled up version of the other so scaled scaled up or version of the other down versions when we talked about congruence they have to be exactly the same you can rotate it you could shift it you could flip it but when you do all those things they would have to essentially be identical with similarity you can rotate it you can shift it you can flip it and you can also scale it up and down in order for something to be similar so for example if you say if something is congruent if if let me say triangle let's say triangle CDE if we know that triangle CDE is congruent to triangle fgh then we definitely know that they are similar they are scaled up by a factor of 1 then we know for a fact that CDE is also similar to triangle fgh but we can't say it the other way around if if triangle ABC is similar to XYZ we can't say that it's necessarily congruent and we see for this particular example they definitely are not congruent so this is one way to think about similarity the other way to think about similarity is that all of the corresponding angles will be equal so if something is similar then all of the corresponding angles are going to be congruent corresponding corresponding angle I always have trouble spelling this it is two R's 1s corresponding corresponding angles corresponding angles are congruent are congruent so if we say that triangle ABC is triangle ABC is similar to triangle XYZ that is equivalent to saying that angle angle ABC angle ABC is congruent or we can say that their measures are equal to angle XYZ to angle X Y Z that angle that angle BAC B AC is going to be congruent to angle yxz to angle Y x z and then finally angle ACB ACB is going to be congruent to angle to angle XY is XZ y X Z Y angle X Z Y so if you have to add two triangles all of their angles are the same then you could say that they're similar or if you find two triangles and you're told that they are similar triangles and you know that all of their corresponding angles are the same and the last I guess way to think about it is that the sides are all just scaled up versions of each other so the sides so sides scaled by the same factor scaled by same same factor in the example we did here the scaling factor was three doesn't have to be three just has to be the same scaling factor for every side if this side if we scale this side up by three and we only scale this side up by two then we would not be dealing with a similar triangle but if we scaled all of these sides up by seven that that's still a similar as long as you have all of them scaled up by or scale down by the exact same factor so one way to think about it is and I want to keep having well I want to still visualize those triangles let me let me redraw them right over here a little bit simpler because I'm not talking in now in general terms not even for that specific case so if we say that this is a B and C and this right over here is x x y and z I just redrew them so I can refer to them when we write down here if we're saying that these two things right over here are similar that means that corresponding sides are scaled up versions of each other so we could say that the length of a B we could say that the length of a be a B is equal to some scaling factor and this thing could be less than one some scaling factor times the length of XY the corresponding sides and I know that a B corresponds to XY because of the order in which I wrote this similarity statement so some scaling factor times XY we know that BC the length of BC we know the length of BC needs to be that same scaling factor that same scaling factor times the length of Y Z times the length of Y Z so that same scaling factor and then we know the length of AC the length of AC is going to be equal to that same scaling factor times X Z so that's X Z and this could be a scaling factor so if a B is larger than if ABC is larger than XYZ then these caves will be larger than one if they're the exact same size if they're essentially congruent triangles then these K's will be one and if X Y Z is bigger than ABC then these Kaling factors will be less than one but another way to write the same statements notice all I'm saying is corresponding sides or scaled up versions of each other this first statement right here if you divide both sides by X Y you get a B over X Y is equal to our scaling factor and then the second statement right over here if you divide both sides by Y Z you get B let me do that same color you get BC divided by Y Z is equal to that scaling factor is equal to a scaling factor remember the example we just showed that scaling factor was scaling factor was three but now we're saying in the more general term similarity as long as you have the same scaling factor and then finally if you divide both sides here by the length between X ends X Z or segment X is YZ length you get a C over X Z is equal to K as well is equal to K as well or another way to think about it is the ratio between corresponding sides notice this is the ratio between a B and X Y a B and XY the ratio between BC and Y Z BC and y Z the ratio between AC and XZ AC and X see that the ratio between corresponding sides all gives us the same constant or you can write rewrite this as a B over XY is equal to BC over Y Z is equal to is equal to AC over rxz which would be equal to some scaling factor which is equal to K so if you have similar triangles let me draw an arrow right over here similar triangles means that they're scaled up versions and you can also flip and rotate and do all this stuff with congruence and you can scale them up or down which means all of the corresponding angles are congruent which also means that the ratio between corresponding sides is going to be the same constant for all the corresponding sides or the ratio between corresponding sides is constant